ANOTHER PROOF OF A RESULT OF N. J. KALTON, E. SAAB
AND P. SAAB ON THE DIEUDONNE PROPERTY IN C(K, E)
by G. EMMANUELEt
(Received 27 August, 1987)
1. Let K be a compact Hausdorff topological space and £ be a Banach space not
containing Z1. Recently N. J. Kalton, E. Saab and P. Saab ([5]) obtained the results that
under the above assumptions the usual space C{K, E) has the Dieudonn6 property; i.e.
each weakly completely continuous operator on C(K, E) is weakly compact. They use
topological results concerning multivalued mappings in their proof. In this short note we
furnish a new and simpler proof of that result without using topological results but only
well known theorems of Bourgain ([2]) and Talagrand ([8]) on weak compactness of sets
of Bochner integrable functions; i.e. results in vector measure theory. At the end of the
paper we present some applications of the result to Banach spaces of compact operators.
We take this opportunity to thank very much Prof. Paula Saab who suggested the
submission of the present proof of her result.
2. In order to give our proof we need a definition and three lemmata. The first one
is due to C. Fierro Bello ([4]).
DEFINITION (See [1].) Let T:C(K, £ ) - » F be an operator. Then there is a vector
measure m from Bo{K) into B(E, F) such that
= {f(s)dm
JK
provided Tis weakly completely continuous (where Bo(K) denotes the a-algebra of Borel
subsets of K and B{E, F) the Banach space of operators from E into F). Further it is
known that m has a control measure A defined on Bo(K). Usually m is named the
representing measure of T.
LEMMA 1 ([4]). Let A be a countably additive, positive and finite Borel measure on K.
We denote by cabvA(flo(/Q, E*) the subspace of all the absolutely X-continuous measures
in cab\(Bo(K), E*). If I:C(K, E)->L\X, E) is the canonical embedding and I* the
conjugate operator from (L\X, £))* into cabv(Bo(/C), E*) then /*(L1(A, £))*) is dense
in cabw^Bo(K), E*).
Moreover we need the other two results contained in Lemma 2 and Lemma 3 in
which T, m and A are as in the Definition above.
LEMMA 2. Let (gn) be a sequence in L'(A, E) such that sup sup ||gn(.s)|| <°° with
" s
gn-^ 6 in L\k, E). Then one has jKgn(s) dm^^Q in F.
t Work performed under the auspices of G.N.A.F.A. of C.N.R. and partially supported by M.P.I, of Italy.
Glasgow Math. J. 31 (1989) 137-140.
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138
G. EMMANUELE
Proof. First of all we observe that $K8n(s) dm makes sense, for each n e N, because
each gn is A-measurable and bounded. Now using Lusin's Theorem ([3]) we can construct
a sequence (Kn) of compact subsets of K such that (i) Kn a Kn+1, (ii) gn restricted to Kn
(in symbols g n |^ n ) is continuous, (iii) the sequence (k(K — Kn)) converges to zero (and
then also the sequence (\\m(K - Kn)\\) converges to zero). The Borsuk-Dugundji's
Theorem ([7]) allows us to define, for each n e N, an extension operator Sn: C(Kn, E)-*
C{K,E) with \\Sn\\ = l. We have that Sn(gn\Kn)(s) = gn(s) for all seKn. Putting
fn = Sn(gn\Kn) we get (/„ -gnXs)-^
d a.e. in K. From the inequality
max(supsup||/ n (s)||, supsup ||g n (s)||)<°° we easily get | | / n - g n | | i - > 0 . Thus we have
/ n - ^ 0 in L\X,E).
If x*eF* we obtain (m, x*) ecab\x(Bo(K), E*) and by
virtue of Lemma 1 $Kfn(s) dm-^
6 in F. On the other hand one has
/»(*) -gn(s)dm\\« sup sup \\fn(s) -gn(s)\\ \\m{KII
(
stKn
Kn)\\^0
n
from which our result follows.
LEMMA
3. Let (hn) be a sequence in Ll(X, E) such that sup sup ||/in(s)|| <<» and
n
s
(hn(s)) is a weak Cauchy sequence for all s e K. Then there is z e F such that
Proof. Let (EP) be a sequence of positive numbers with S £P < °°. We can construct a
sequence (Kp) of compact subsets of K such that (i) Kp<zKp+x, (ii) hn\Kp = %pnp is
continuous for each n,peN, (iii) \\m(K- Kp)\\ <ep for each p e N. Using again the
Borsuk-Dugundji's theorem we can define the following functions q>np = sp(%i>np) for each
n,p efi. Of course (<pnp)n is a weak Cauchy sequence in C(K, E) for each p eN. Then
for allp e N there is a zp eFfor which T(q>np)-^+ zp. Now we observe that
II?>„.,(*) - <Pn.P+i{s)\\ dm =£ const. ep
lim
n
JK
where the constant doesn't depend on s e K, n, p e N. Hence the sequence (zp) must
converge to some z eF. It is then easy to see that (T(q>nn)) converges weakly to z. On
the other hand we have
n(s)
~ <Pn,n(S) dm\\ =
h
n(i) ~ Vn,n(S) dm\ « COnSt. En - * 0.
K-K.
Hence we get jKhn(s)
dm-^z.
The proof is complete.
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THE DIEUDONNE PROPERTY IN C(K, E)
139
Now we are able to present our proof of the result in [5].
Let E be a Banach space not containing I1. Then C(K, E) has the
Dieudonne property.
THEOREM.
Proof. Let (/„) be a bounded sequence in C(K, E). The sequence (/(/„)) = (/„) is
uniformly integrable in L'(A, E). A result of Bourgain ([2]) gives that (/„) is conditionally
weakly compact. We may suppose that it is a weak Cauchy sequence; otherwise we pass
to a subsequence. Using results by Talagrand ([8]) we have that /„ can be written as the
sum of two functions gn and hn for each n e N and a.e. in K. Further from the cited results
of [8] it follows that (#„) converges weakly to 6 and (hn) may be chosen so that (hn(s)) is
weak Cauchy for each s e K. An inspection of the proof of those results also gives that
sup sup ||gn(s)|| <°°, sup sup P R (s)ll <°° since this is true for (/„). Having
f fn(s) dm = f gn(s) dm + f hn(s) dm
JfC
->K
JK
we get our thesis by virtue of Lemma 2 and Lemma 3.
The Theorem above has the following consequences relative to Banach spaces of
compact operators. If X and Y are two Banach spaces, KW.{X*, Y) denotes the Banach
space of weak* weakly continuous and compact operators from X* into Y equipped with
the operator norm ([6]).
1. Let X be injective and let Y not contain a copy of I1. Then KW.(X*, Y)
has the Dieudonne property.
COROLLARY
Proof. KW.(X*, Y) is isomorphic to KW.(Y*, X) which is complemented in
KW'(Y*, C(BX')) in an obvious way. {BX' denotes the unit dual ball of X.) This last space
is isomorphic to C{BX', Y) (see [6]). An application of the Theorem concludes the proof.
In the following corollary K(X, Y) denotes the Banach space of compact operators
from X into Y equipped with the operator norm.
2. Let Z be an i^-space and Y not containing I1. Then K(Z*, Y) has the
Dieudonne property.
COROLLARY
Proof. K(Z*, Y) is isomorphic to KW.(Z***, Y) ([6]). Since it is well known that
Z** is injective an appeal to Corollary 1 concludes the proof.
REFERENCES
1. J. Batt, G. Berg, Linear bounded transformations on the space of continuous functions, J.
Functional Analysis 4 (1969), 215-239.
2. J. Bourgain, An averaging result for /' sequences and applications to weakly conditionally
compact subsets in Vx, Israel]. Math. 32 (1979), 289-298.
3. N. Dinculeanu, Vector measures (Pergamon Press, 1967).
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at https://www.cambridge.org/core/terms. https://doi.org/10.1017/S0017089500007655
140
G. EMMANUELE
4. C. Fierro Bello, On weakly compact and unconditionally converging operators in spaces of
vector valued continuous functions, preprint.
5. N. J. Kalton, E. Saab, P. Saab, On the Dieudonne property for C(K, E), Proc. Amer.
Math. Soc. 96 (1986), 50-52.
6. W. Ruess, Duality and geometry of spaces of compact operators, Math. Studies 90 (North
Holland, 1984).
7. Z. Semadeni, Banach spaces of continuous functions (P. W. N. Warszawa, 1971).
8. M. Talagrand, Weak Cauchy sequences in L\E), Amer. J. Math. 106 (1984), 703-724.
DEPARTMENT OF MATHEMATICS
UNIVERSITY OF CATANIA
VIALE A. DORIA 6
CATANIA 95125
ITALY
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